LMFDB - L-function 2-6048-8.5-c1-0-65

Dirichlet series

L(s) = 1

+ 3.46i·5^-s − 7^-s + 3.31i·11^-s + 3.10i·13^-s − 1.40·17^-s − 4.80i·19^-s − 8.79·23^-s − 6.99·25^-s − 9.87i·29^-s + 7.83·31^-s − 3.46i·35^-s + 5.42i·37^-s − 11.5·41^-s − 7.03i·43^-s − 11.2·47^-s + ⋯

L(s) = 1

+ 1.54i·5^-s − 0.377·7^-s + 1.00i·11^-s + 0.862i·13^-s − 0.340·17^-s − 1.10i·19^-s − 1.83·23^-s − 1.39·25^-s − 1.83i·29^-s + 1.40·31^-s − 0.585i·35^-s + 0.891i·37^-s − 1.80·41^-s − 1.07i·43^-s − 1.64·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$-0.0841 + 0.996i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (3025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ -0.0841 + 0.996i)$

Particular Values

$L(1)$	$\approx$	$0.1284181636$
$L(\frac12)$	$\approx$	$0.1284181636$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + T $
good	5	$ 1 - 3.46iT - 5T^{2} $
	11	$ 1 - 3.31iT - 11T^{2} $
	13	$ 1 - 3.10iT - 13T^{2} $
	17	$ 1 + 1.40T + 17T^{2} $
	19	$ 1 + 4.80iT - 19T^{2} $
	23	$ 1 + 8.79T + 23T^{2} $
	29	$ 1 + 9.87iT - 29T^{2} $
	31	$ 1 - 7.83T + 31T^{2} $
	37	$ 1 - 5.42iT - 37T^{2} $
	41	$ 1 + 11.5T + 41T^{2} $
	43	$ 1 + 7.03iT - 43T^{2} $
	47	$ 1 + 11.2T + 47T^{2} $
	53	$ 1 - 6.51iT - 53T^{2} $
	59	$ 1 - 3.89iT - 59T^{2} $
	61	$ 1 + 9.42iT - 61T^{2} $
	67	$ 1 - 0.909iT - 67T^{2} $
	71	$ 1 - 6.42T + 71T^{2} $
	73	$ 1 - 1.56T + 73T^{2} $
	79	$ 1 - 11.1T + 79T^{2} $
	83	$ 1 - 0.370iT - 83T^{2} $
	89	$ 1 - 4.85T + 89T^{2} $
	97	$ 1 - 1.63T + 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−7.80509584684739649973320202987, −6.94976305775526383474054784198, −6.58370666543008753645378767009, −6.11465471235099183549130646197, −4.85004619147125529753434033495, −4.19270096931919908117392773903, −3.36770603786770231006670321462, −2.45146797888473917363368237887, −1.95364650396496352615502571638, −0.03490738073083201860118625796, 1.00077602996809769598294945907, 1.89377963050074349213719889046, 3.21615100795973700706195336373, 3.80330426584092474486455147774, 4.78042440658428065212047951347, 5.36990009634371323158104395928, 6.02073519200834462002136131662, 6.69172438619381469705982458802, 8.085653385735728000510070797645, 8.099248021732456278748026967184

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1284181636\)
\(L(\frac12)\)	\(\approx\)	\(0.1284181636\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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